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Shy couplings, CAT(0) spaces, and the lion and man
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Bramson, Maury, 1951, Burdzy, K. (Krzysztof) and Kendall, Wilfrid S.. (2013) Shy couplings, CAT(0) spaces, and the lion and man. Annals of Applied Probability, 41 (2). pp. 744784. ISSN 10505164

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Official URL: http://dx.doi.org/10.1214/11AOP723
Abstract
Two random processes X and Y on a metric space are said to be εshy coupled if there
is positive probability of them staying at least a positive distance ε apart from each other forever.
Interest in the literature centres on nonexistence results subject to topological and geometric conditions; motivation arises from the desire to gain a better understanding of probabilistic coupling.
Previous nonexistence results for coadapted shy coupling of reflected Brownian motion required
convexity conditions; we remove these conditions by showing the nonexistence of shy coadapted
couplings of reflecting Brownian motion in any bounded CAT(0) domain with boundary satisfying uniform exterior sphere and interior cone conditions, for example, simplyconnected bounded
planar domains with C2 boundary.
The proof uses a CameronMartinGirsanov argument, together with a continuity property of
the Skorokhod transformation and properties of the intrinsic metric of the domain. To this end, a
generalization of Gauss' Lemma is established that shows differentiability of the intrinsic distance
function for closures of CAT(0) domains with boundaries satisfying uniform exterior sphere and
interior cone conditions. By this means, the shy coupling question is converted into a Lion and
Man pursuitevasion problem.
Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Brownian motion processes 
Journal or Publication Title:  Annals of Applied Probability 
Publisher:  Institute of Mathematical Statistics 
ISSN:  10505164 
Official Date:  March 2013 
Volume:  41 
Number:  2 
Page Range:  pp. 744784 
Identification Number:  10.1214/11AOP723 
Status:  Peer Reviewed 
Publication Status:  Published 
Access rights to Published version:  Restricted or Subscription Access 
Funder:  National Science Foundation (U.S.) (NSF), Poland. Ministerstwo Nauki i Szkolnictwa Wyższego [Ministry of Science and Higher Education] (MNiSW) 
Grant number:  CCF0729537 (NSF), DMS0906743 (NSF), N N201 397137 (MNiSW) 
References:  Alexander, S. B., R. L. Bishop, and R. Ghrist (2006). Pursuit and evasion in nonconvex domains 
URI:  http://wrap.warwick.ac.uk/id/eprint/37521 
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